Update: had a go with it this evening (air filter off, bolt back in).
Choke plate opens, engine runs rich for 15 seconds or so, then cuts
out. Petrol mist seen in carb throat. No luck from needle valve
adjustments (thanks D. Martel). Governor has to move a long way before
it takes up the slack in the spring I've noticed, and the idle speed
screw doesn't hit the end point. Have emailed local friendly B&S dealer
about carb overhaul parts pricing/availability. It's all so frustrating
as it's a nice bit of kit and too good to get rid of.

Will let you know how I get on (maybe a week or so).
James